"Lay your plane on it's side son."
This is the long-standing advice. It is old, nobody knows how old, but many (most?) now don't do this, some to deliberately buck Old Timer advice (which can often seem folksy and overly cautious*) but some because they've thought through the problem and come to a definite conclusion not to.
The evidence is in and it appears that you don't need to set a hand plane down on its side and I think the majority no longer do this. But it's still worth looking into it a little and see whether it might be good advice, as it apparently once was.
Why you should
The usual explanation given for why you should do this is to prevent damage to the cutter, and at first glance this does seem to make sense. But then you think about it a bit and realise that the edge of this cutter is (repeatedly, and at speed!) going to be asked to run over wood much like you were about to place it down on and you're not worried about it being damaged by that. So it sort of doesn't make sense.
But then you think about it just a little further — a benchtop is not the same as the wood you are working. Its surface is older, maybe scarred from use (look at old benches and their tops are often in horrible shape, even in really good workshops producing quality stuff) and all those imperfections might hold untold horrors when it comes to abrasive particles from many possible sources. I wouldn't want to risk the finely honed edge of a plane iron on that, much less risk picking up some grit on the sole of the plane (lubricated as they were with oil or grease) which could be transferred to the workpiece.
Why you shouldn't
One reason given for why we shouldn't do this now is that the old advice comes from an era when planes were made from wood with an iron very firmly held by a wedge, and so they retained lateral adjustment much better than metal bench planes with the usual mechanism. So, the thinking goes, any of the standard Bailey-style metal planes may lose lateral adjustment if laid on their side.
I find this superficial thinking and unsatisfying because of course many generations of users of metal planes were following the old advice by rote and dutifully laying their planes on their side and obviously weren't experiencing problems with lateral adjustment being thrown off. We can know this with some certainty because if it had been occurring regularly we'd have heard about the risk through the anecdotal sources and it would be immortalised in print in at least some of the books, and yet both are silent on this possibility. Well, until now and the easy spread of oddball ideas through the Internet!
Another reason given is that a plane laid on its side exposes the sole with the iron projecting slightly and this poses a danger to knuckles etc. accidentally bumped into or passing by the plane as you reach for something. This is much more convincing. I've nearly cut myself in just this way so it's not merely a theoretical danger.
The bottom line for a lot of us
More than a few current woodworking gurus using hand planes, with both wood and metal bodies, don't lay their planes on their sides and haven't done for many decades and have never had an iron's edge become damaged as a result.
Since most of us will be using our planes far, far less then they will and in a similar working environment (clean, or at least cleanish, indoor workshops) it's a fairly safe bet that you can safely follow suit.
However, if you work on building sites and the surface you lay your plane down on might be contaminated with who-knows-what dust, metal particles and random specs of grit from many possible sources then it only makes sense to be more cautious and lay your planes on their sides.
Working at home you can follow the old advice and lay your planes on their sides if you want to, regardless of your reasoning, but I would caution to lay them down the the sole facing away from the side(s) of the bench you commonly stand at. Or lay it in your tool tray if you have one.
*You'll read in some older guides that chisels were supposed to be placed bevel down as well to protect their edges, which most assuredly seems unnecessarily cautious to modern users working in their home workshops.